compass and straightedge construction of perpendicular

Let $P$ be a point and $\ell$ be a line in the Euclidean plane. One can construct a line $m$ perpendicular to $\ell$ and passing through $P$ . The construction given here yields $m$ in any circumstance: Whether $P \in \ell$ or $P \notin \ell$ does not matter. On the other hand, the construction looks quite different in these two cases. Thus, the sequence of pictures on the left (in which $\ell$ is in red) is for the case that $P \notin \ell$ , and the sequence of pictures on the right (in which $\ell$ is in green) is for the case that $P \in \ell$ .

1. With one point of the compass on $P$ , draw an arc that intersects $\ell$ at two points. Label these as $Q$ and $R$ .
   
   
   
2. Construct the perpendicular bisector of $\overline{QR}$ . This line is $m$ .
   
   
   

This construction is justified because $Q$ and $R$ are constructed so that $P$ is equidistant from them and thus lies on the perpendicular bisector of $\overline{QR}$ .

In the case that $P \notin \ell$ , this construction is referred to as dropping the perpendicular from $P$ to $\ell$ . In the case that $P \in \ell$ , this construction is referred to as erecting the perpendicular to $\ell$ at $P$ .

If you are interested in seeing the rules for compass and straightedge constructions, click on the link provided.